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Introduction to the theory of functions with generalized derivatives, and quasiconformal mappings. (Vvedenie v teoriyu funktsij s obobshchennymi proizvodnymi i kvazikonformnye otobrazheniya). (Russian) Zbl 0591.46021
Moskva: ”Nauka”. 285 p. R. 3.10 (1983).
In the 1930s S. L. Sobolev started his pioneering work on differentiation of ”nonsmooth” functions. He succeeded in creating a mathematically rigorous theory of differentiation of nonsmooth functions, which enables him to study partial differential equations with nonsmooth solutions. These revolutionary ideas have had a great impact on the development of the mathematical analysis in this century (PDE theory, calculus of variations, differential geometry, and nonlinear potential theory). We mention a few works related to the potential-theoretic aspect of Sobolev spaces \(W^ p_ n:\) the books of V. G. Maz’ya [Einbettungssätze für Sobolevsche Räume. I (1979; Zbl 0429.46018), II (1980; Zbl 0438.46021)] and of B. W. Schulze and G. Wildenhain [Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordnung (1977; Zbl 0369.35001)] and the survey of J. Frehse [Jahresber. Deutsch. Math.-Verein. 84, 1-44 (1982; Zbl 0486.35002)].
The book of Gol’dshtein and Reshetnyak under review deals with some interesting novel topics of Sobolev spaces. It is self-contained and consists of six chapters, as follows. Chapter I: Introduction (including the Radon-Nikodym theorem and Vitali’s covering lemma). Chapter II: Functions with generalized derivatives (including integral representation of differentiable functions in domains of class J, estimates for integrals of potential type, \(W^ l_ p\)-classes, embedding theorems and approximation by smooth functions, differentiability a.e. of generalized functions). Chapter III: Nonlinear capacity (including the capacity connected with a positive linear operator, variational capacity, the Hausdorff h-measure and sufficient conditions for zero (l,p)- capacity, estimates for (l,p)-capacity for certain pairs of sets). Chapter IV: The density of extremal functions in the Sobolev space of functions with generalized first derivatives (including extremal functions for (1,p)-capacity, approximation of \(L^ 1_ p\) functions by extremal functions, removable singularities for spaces \(L^ 1_ p(G))\). Chapter V: Change of variables (including the multiplicity and degree of a mapping, a change of variables formula for integration of Sobolev space functions, the invariance of the space \(L^ 1_ n(G)\) under quasi- isometric and quasi-conformal mappings). Chapter VI: Extension of Sobolev space functions (including necessary conditions for extension, and also sufficient conditions). The treatment of several topics is influenced by the authors’ original research. Many results, e.g., the embedding theorem, are proved for a very wide class (class J) of domains, which is substantially larger than the class of starshaped domains. The class J of domains was introduced by F. John and it was considered by Reshetnyak in his papers. Chapter V relates Sobolev spaces and quasiconformal mappings to each other. Chapter VI deals with some remarkale recent results of Gol’dshtein, S. K. Vodopyanov and P. W. Jones on extension operators of Sobolev spaces.
The booklet of Gol’dshtein alone (bibliographical data in Zbl 0591.46022 below) is intended to be a textbook, giving an introduction to some of the material expounded in the larger book. It contains three chapters: I. Capacity induced by a linear operator, II. Variational capacity and embedding theorems in the space of continuous functions, III. Continuation of functions across the boundary of their domain of definition.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
30C62 Quasiconformal mappings in the complex plane